Quantum thermal machine regimes in the transverse-field Ising model

We consider a one-dimensional working substance consisting of $N$ spin-1/2 particles with nearest-neighbour interactions. The Hamiltonian of the system is given by the transverse-field Ising model,

where $\hat{\sigma}_{j}^{\mu}$ are Pauli operators acting on the $j^{th}$ site ($[\hat{\sigma}_{j}^{\mu},\hat{\sigma}_{j}^{\nu}]=2i\epsilon_{\mu\nu\kappa}\hat{\sigma}_{j}^{\kappa}$), $g>0$ is the interaction strength and $h$ is the transverse field strength. We impose periodic boundary conditions $\hat{\sigma}_{N+1}=\hat{\sigma}_{1}$. In the thermodynamic limit ($N\rightarrow\infty$) the transverse-field Ising model has a quantum critical point at $g=|h|$ separating the ferromagnetic ($g>|h|$) and paramagnetic ($g<|h|$) ground states [38]. Equation (1) can be diagonalised following a Jordan-Wigner transformation from spins to free fermions. For large $N$ this gives [39]

with $\hat{c}_{j}$ the fermion annihilation operators for modes $j=1,...,N$. The fermionic mode energies are $\omega(k_{j})$ with $k_{j}=\pi(2j-1)/N$ and

The ground-state energy is $E_{0}=-\frac{1}{2}\sum_{j=1}^{N}\omega(k_{j})$.

Among the various possible thermodynamic cycles that can be used to realise thermal machines, we choose an Otto cycle due to the ease with which heat and work can be distinguished [40, 41]. The quantum Otto cycle consists of four strokes as illustrated in Fig. 1(a) [42]. The system begins in a hot thermal state at 1. Step (1 $\rightarrow$ 2): The system is thermally isolated and the driving field $h$ is adiabatically tuned from $h_{H}$ to $h_{C}$. Work $W_{1\rightarrow 2}$ is done by the system. Step (2 $\rightarrow$ 3): The system is coupled to the cold reservoir with $h$ fixed, exchanging heat $Q_{C}$ until thermal equilibrium is achieved at 3. Step (3 $\rightarrow$ 4): The system is thermally isolated and $h$ is adiabatically tuned from $h_{C}$ to $h_{H}$. Work $W_{3\rightarrow 4}$ is done by the system. Step (4 $\rightarrow$ 1): The system is coupled to the hot reservoir with $h$ fixed, exchanging heat $Q_{H}$ until thermal equilibrium is achieved at 1.

The total work done in the cycle is

The first and second laws of thermodynamics permit four possible regimes of thermal machines, depending on the signs of heat and work [43], see Fig. 1(b):

• •

  Engine [E]: Utilises the thermodynamic flow of heat to do work; $W<0,Q_{H}>0,Q_{C}<0$.

• •

  Accelerator [A]: Increases the thermodynamic flow of heat using work; $W>0,Q_{H}>0,Q_{C}<0$.

• •

  Refrigerator [R]: Utilises work to reverse the thermodynamic flow of heat; $W>0,Q_{H}<0,Q_{C}>0$.

• •

  Heater [H]: Utilises work to transfer heat to both reservoirs; $W>0,Q_{H}<0,Q_{C}<0$.

We use the convention that a negative value of heat or work is an output and a positive value an input. The thermodynamic flow of heat is a flow of heat from the hot to the cold reservoir.

In the thermodynamic limit the spacing $k_{j+1}-k_{j}$ between fermionic modes becomes infinitesimal, and summations $\sum_{j}$ over fermionic modes can be replaced by an integral $\frac{N}{\pi}\int_{0}^{\pi}dk$. The total adiabatic work output for the cycle in Fig. 1 is then [22]

Here $f_{H,C}(k)\equiv[1+e^{\omega_{H,C}(k)/k_{B}T_{H,C}}]^{-1}$ is the Fermi-Dirac distribution, which gives the occupation of the fermionic modes in thermal equilibrium.

In general, the thermal machine regime is determined by how energy levels respond to changes in $h$ and the number of energy levels contributing (i.e. temperature). To simplify our initial analysis, we consider an infinitesimal work stroke $\delta h=h_{H}-h_{C}$. For $\delta h\rightarrow 0$, Eq. (5) simplifies to

Here $\langle\cdot\rangle_{T}$ denotes an expectation value with respect to a thermal state at temperature $T$. The quantity $\delta h\left<\frac{d\hat{H}}{dh}\right>_{T}$ is the change in system energy due to the infinitesimal work stroke $\delta h$ acting on a thermal state at temperature $T$. Similarly, the heat associated with the hot and cold reservoirs are,

For $\delta h\rightarrow 0$ we have $Q_{H}>0$ and $Q_{C}<0$ and hence only accelerator or engine operation is possible, depending on the sign of work. We also have the relation:

where $F=-k_{B}T\ln Z_{T}$ is the free energy of the system, with $Z_{T}=\operatorname{Tr}e^{-\hat{H}/k_{B}T}$ the partition function at temperature $T$. Here the advantage of using an infinitesimal work stroke is clear, as it allows work and heat to be computed directly from the free energy. Using Eq. (2) the free energy is then [39]

Furthermore, using the partition function, we have

Here $M(T)=\sum_{j=1}^{N}\langle\hat{\sigma}_{j}^{z}\rangle_{T}$ is the average transverse magnetization of the system (below we will also use the magnetization per particle $m=M/N$). Hence [22, 44]

i.e. the net work exchanged is directly proportional to the difference in transverse magnetization of the system at temperatures $T_{H}$ and $T_{C}$.

Equation (11) takes an analogous form to the work output of an ideal gas Otto cycle, which for infinitesimal volume change $\delta V$ is

with $P=-\frac{dF}{dV}$ the pressure of the system. This pressure is monotonic with temperature, resulting in operation fixed by the sign of $\delta V$. Similarly, setting $g=0$ in Eq. (1) results in operation fixed by the sign of $\delta h$. In general, for simple non-interacting systems where all energy levels vary monotonically with the control parameter, the type of operation is fixed by the sign of the change in control parameter.

The work output and regimes of operation for the Otto cycle (Fig. 1) with infinitesimal work step $\delta h>0$ are shown in Fig. 2(a). As already noted, with infinitesimal work step only accelerator or engine regimes are possible, with the regime of operation determined by the sign of $M(T_{H})-M(T_{C})$ (Eq. (11)). When $M(T_{H})<M(T_{C})$ the system operates as an engine, whereas for $M(T_{H})>M(T_{C})$ the system operates as an accelerator. For $\delta h<0$, the sign of work is reversed and the engine and accelerator regimes are switched.

Due to the connection between work and $M(T)$, the regimes of operation can be understood from the behaviour of $M(T)$. Example behaviour of $M(T)$ are illustrated in Fig. 2(b) and Fig. 3. The engine and accelerator regimes are separated by a boundary where $W=0$ and $M(T_{H})=M(T_{C})$, which requires non-monotonic behaviour in $M(T)$, see Fig. 2(b)(iii).

We now briefly discuss the magnitude of work extracted (engine) or consumed (accelerator). In the high-temperature, low-$h$ limit, $|W|$ increases with increasing $h$, see Fig. 2. This can be seen directly from the high temperature expansion Eq. (19), which also results in $|M(T_{H})-M(T_{C})|$ decreasing with increasing $T$ for fixed $T_{H}/T_{C}$. In contrast, in the low-temperature, high-$h$ limit, $|W|$ decreases with increasing $h$, see Fig. 2. When $h\gg g$, the transverse magnetization can be approximated by the magnetization of non-interacting spins,

For low temperatures $k_{B}T\ll h$ this gives $M\approx N(1-2e^{-h/k_{B}T})$, in which case we find $|M(T_{H})-M(T_{C})|$ increases with increasing $T$ for fixed $T_{H}/T_{C}$. Both of these arguments also hold in the absence of interactions.

For $k_{B}T\lesssim g$, $|M(T)-M(0)|$ is largest around $h\approx g$, see Fig. 5. This results in a peak in $|W|$ for $h\approx g$, see Fig. 2. Such a peak has been discussed already in the low-temperature regime, in which case it can be explained in terms of a decreasing energy gap between the ground and first excited state [16]. At low temperatures and for $h>g$, this results in a performance that exceeds that of non-interacting spins [16].

For an infinitesimal work stroke, the sign of the heat flows are fixed, restricting operation to either engine or accelerator, see Eq. (7). A finite-sized work stroke $\Delta h=h_{H}-h_{C}$ permits the thermal machine to operate as a refrigerator or a heater, see Fig. 6, where we have defined $h_{\mathrm{av}}=(h_{H}+h_{C})/2$.

To see how refrigeration operation may appear, it is simplest to first consider very low temperatures, in which case only the ground and first excited states have any significant thermal occupation [16]. The analysis then proceeds as for a single spin-1/2 system [43]. For low temperatures the heat flows are,

with $|h-g|$ the energy of the first excited level, which is the $k=0$ fermionic mode. From examination of Eq. (25), it is clear that low-temperature refrigeration ($Q_{H}<0$, $Q_{C}>0$) occurs when

For $\Delta h>0$ Eq. (26) is satisfied when

The bounds of the inequality (27) accurately predict the low-temperature refrigerator boundary, see Fig. 6(b). Similar inequalities can be obtained from Eq. (26) for $\Delta h<0$.

For higher temperatures, the analysis is qualitatively similar but complicated by the many levels present in the problem. This case has been discussed in recent works [37, 34, 22]. Notably, refrigerator operation is more effective when the temperature difference between $T_{H}$ and $T_{C}$ is small, and the influence of the critical point results in a peak in cooling capability [37]. Additionally, it was shown that as the size of the work stroke increases, the boundary of the refrigerator region expands to smaller $T_{C}$ [34]. This expansion of the refrigerator region becomes more pronounced with work strokes across the critical point, as seen in Fig. 6.

All four regimes of operation appear for finite $\Delta h$ at sufficiently high temperature, see Fig. 6(a). This gives rise to a “Carnot point” where $Q_{H}=Q_{C}=W=0$ and all four regimes intersect [43, 22]. A sufficient condition for $Q_{H}=Q_{C}=W=0$ is $r(k)=\omega_{H}(k)/\omega_{C}(k)=T_{C}/T_{H}$ for all $k$ [22]. The ratio $r$ is insensitive to $k$ when $dr(k)/dk=0$, which gives $g^{2}=h_{H}h_{C}$ [using Eq. (3)]. Substituting this into $r$ and setting $r=T_{C}/T_{H}$ gives values $h_{C}^{\mathrm{cp}}$ and $h_{H}^{\mathrm{cp}}$ where $Q_{H}=Q_{C}=W=0$ for given $T_{C}$, $T_{H}$ and $g$ [22],

The field strengths $h_{H}^{\mathrm{cp}}$ and $h_{C}^{\mathrm{cp}}$ are shown in Fig. 7. The Carnot point appears when $\Delta h$ crosses $g$, since $h_{C}^{\mathrm{cp}}<g<h_{H}^{\mathrm{cp}}$. This is reasonable because the sign of $\frac{d\omega}{dh}$ is most sensitive to $h$ near $g$, so small variations in $h_{\mathrm{av}}$ can lead to changes in the sign of $W,Q_{H}$ and $Q_{C}$. For small $\Delta h$, the Carnot point will appear when $T_{H}\approx T_{C}$, as then only a small $\Delta h$ is needed to change the flow of heat, see Fig. 6(a)(ii). As $\Delta h$ increases ($h_{H}\gg g$), the accelerator and heater regimes shrink, and the system behavior converges to that of a noninteracting spin chain, where only the engine and refrigerator regimes remain.

When $k_{B}T_{H}-k_{B}T_{C}\gg|\Delta h|$ the sign of the heat flows are fixed and only accelerator or engine operation is possible. Furthermore, the $W=0$ boundary separating these two regimes is unaffected by $\Delta h$ at low temperatures, see Fig. 6. Focusing on just the ground and first excited state, as in Eq. (25), zero work output will occur when $|h_{C}-g|=|h_{H}-g|$, in which case the work stroke causes no net change in the energy of the first excited state. This gives $h_{\mathrm{av}}=g$ irrespective of $\Delta h$.